Forecasters say snow showers or longer periods of snow are expected on Tuesday, accompanied by very low temperatures, across much of Wales.
There is the potential for travel delays on roads, with some stranded vehicles and passengers, as well as delays or cancellations to rail and air travel.
Some rural communities could become cut off. Power cuts may also occur and other services, such as mobile phones, may be affected.
Met Office meteorologist Charlie Powell said:
"The UK is on track for some really cold weather this week. It's not going to be record-breaking, but it'll be pretty exceptional - winds are going to make it feel minus 10C to minus 15C in the day."
We will see the first signs of that tonight in the shape of snow showers working all the way down the east coast. That continues into Monday, with snow showers moving across the country during the day before reaching Wales."
Snow has already fallen in other parts of the UK including London that woke up to a flurry on Monday morning.